有限域上的块伴随Singer环、本原递归向量序列和互质多项式对
Block Companion Singer Cycles, Primitive Recursive Vector Sequences, and Coprime Polynomial Pairs over Finite Fields
摘要：
本文讨论有限域上块伴随非奇异矩阵计数的一个猜想，这些矩阵的阶在相应的一般线性群中是最大可能的.利用最近的一些结果证明了一种特殊情况下，一对多项式的系数在有限域是互质的概率。与Niederreiter关于给定维数的分裂子空间的数目的一个较老的问题的连接被概述并且建立了一个渐近版本的结构公式。本文还讨论了它在有限域上给定大小的非奇异Toeplitz矩阵计数中的应用。
英文摘要：
We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed.
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